Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building

TL;DR

This paper explores how embedding invariants of field extensions in simple algebras relate to the structure of Euclidean buildings, connecting algebraic invariants with geometric and combinatorial properties.

Contribution

It establishes a relationship between embedding invariants and the simplicial structure of Euclidean buildings associated with central simple algebras over local fields.

Findings

Relation between embedding invariants and simplicial structures

Behavior of the map j_E with respect to Euclidean buildings

Connection between algebraic embeddings and geometric structures

Abstract

This article answers a question that naturally arises from the articles by Grabitz and Broussous "Pure elements and intertwining classes of simple strata in local central simple algebras" and Broussous and Lemaire "Buildings of GL(m,D) and Centralizers". For an Azumaya-Algebra A over a non-Archimedean local field F, Grabitz and Broussous have introduced embedding invariants for field embeddings, that is for pairs (E,a), where E is a field extension of F in A, and $a$ is a hereditary order which is normalised by E^x. On the other hand if we take such a field extension E and define B to be the centralizer of E in A, then G:=A^x and G_E:=B^x are sets of rational points of reductive groups defined over F and E respectively. Broussous and Lemaire have defined a map j_E: I^{E^x}\to I_E, where $I$ is the the Euclidean building of $G$, and I_E is the Euclidean building of G_E. The question which we address is to relate the embedding invariants to the behavior of the map j_E with respect to the simplicial structures of $I$ and I_E. I have to thank very much Prof. Zink from Homboldt University Berlin for his helpful remarks, the revision of the work and for giving my the interesting task.